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Journal of Mathematical Chemistry

, Volume 57, Issue 1, pp 149–179 | Cite as

Numerical simulation for computational modelling of reaction–diffusion Brusselator model arising in chemical processes

  • Sanjay Kumar
  • Ram JiwariEmail author
  • R. C. Mittal
Original Paper
  • 194 Downloads

Abstract

The main focus of this article is to capture the patterns of reaction–diffusion Brusselator model arising in chemical processes such as enzymatic reaction, formation of turing patterns on animal skin, formation of ozone by atomic oxygen through a triple collision. For this purpose, a meshfree algorithm is developed based on radial basis multiquadric functions and differential quadrature (DQ) technique. The algorithm is more general than (Alqahtani in J Math Chem 56:15431566, 2018) due to meshfree and \(C^{\infty }\) properties of radial basis functions. Numerical experiment section support the accuracy and efficiency of the algorithm. The computed results satisfy the theory of Brusselator model which says for small values of diffusion coefficient, the steady state solution converges to equilibrium point \(\left( \alpha , \dfrac{\beta }{\alpha }\right) \) if \(1-\beta +\alpha ^{2}>0\) .

Keywords

Brusselator model Oscillating reactions Mesh free algorithm Stability analysis Numerical experiments 

Notes

Acknowledgements

The work is supported by Science and Engineering Research Board (SERB) with Grant No. SERB/YSS/2015/000599.

References

  1. 1.
    R. Lefever, G. Nicolis, Chemical instabilities and sustained oscillations. J. Theor. Biol. 30, 267 (1971)CrossRefGoogle Scholar
  2. 2.
    G. Nicolis, I. Prigogine, Self-Organization in Non-equilibrium Systems (Wiley, New York, 1977)Google Scholar
  3. 3.
    I. Prigogine, R. Lefever, Symmetries breaking instabilities in dissipative systems II. J. Phys. Chem. 48, 16951700 (1968)CrossRefGoogle Scholar
  4. 4.
    J. Tyson, Some further studies of non-linear oscillations in chemical systems. J. Chem. Phys. 58, 3919 (1973)CrossRefGoogle Scholar
  5. 5.
    R.J. Field, E. Koros, R.M. Noyes, Oscillations in chemical systems. 2. Thorough analysis of temporal oscillation in bromate-cerium-malonic acid system. J. Am. Chem. Soc. 94, 8649–8664 (1972)CrossRefGoogle Scholar
  6. 6.
    P.A. Zegeling, H.P. Kok, Adaptive moving mesh computations for reactiondiffusion systems. J. Comput. Appl. Math. 168, 519528 (2004)CrossRefGoogle Scholar
  7. 7.
    E.H. Twizell, A.B. Gumel, Q. Cao, A second-order scheme for the Brusselator reactiondiffusion system. J. Math. Chem. 26, 297316 (1999)CrossRefGoogle Scholar
  8. 8.
    R.C. Mittal, R. Jiwari, Numerical study of two-dimensional reactiondiffusion Brusselator system. Appl. Math. Comput. 217(12), 54045415 (2011)Google Scholar
  9. 9.
    R. Jiwari, J. Yuan, A computational modeling of two dimensional reactiondiffusion Brusselator system arising in chemical processes. J. Math. Chem. 52, 15351551 (2014)CrossRefGoogle Scholar
  10. 10.
    A.M. Alqahtani, Numerical simulation to study the pattern formation of reaction–diffusion Brusselator model arising in triple collision and enzymatic. J. Math. Chem. 56, 15431566 (2018)CrossRefGoogle Scholar
  11. 11.
    G. Adomian, The diffusion-Brusselator equation. Comput. Math. Appl. 29, 13 (1995)Google Scholar
  12. 12.
    A.M. Wazwaz, The decomposition method applied to systems of partial differential equations and to the reaction–diffusion Brusselator model. Appl. Math. Comput. 110, 251264 (2000)Google Scholar
  13. 13.
    E.H. Twizell’s, A second-order scheme for the “Brusselator” reaction–diffusion system. J. Math. Chem. 26(4), 297–316 (2000)CrossRefGoogle Scholar
  14. 14.
    A.A. Siraj-ul-Islam, S. Haq, A computational modeling of the behaviour of the two-dimensional reaction–diffusion Brusselator system. Appl. Math. Model. 34, 38963909 (2010)CrossRefGoogle Scholar
  15. 15.
    J.G. Verwer, W.H. Hundsdorfer, B.P. Sommeijer, Convergence properties of the Runge–Kutta–Chebyshev Method. Numer. Math. 57, 157–178 (1990)CrossRefGoogle Scholar
  16. 16.
    W.T. Ang, The two-dimensional reaction–diffusion Brusselator system: a dual-reciprocity boundary element solution. Eng. Anal. Bound Elem. 27, 897903 (2003)Google Scholar
  17. 17.
    M. Dehghan, M. Abbaszadeh, Variational multi scale element free Galerkin (VMEFG) and local discontinuous Galerkin (LDG) methods for solving two-dimensional Brusselator reaction–diffusion system with and without cross-diffusion. Comput. Methods Appl. Mech. Eng. 300, 770797 (2016)CrossRefGoogle Scholar
  18. 18.
    T.J.R. Hughes, G.M. Hulbert, A new finite element formulation for computational fluid dynamics: VIII, in The Galerkin/Least Squares Method for Advection Diffusion Equation, Computer Methods in Applied Mechanics and Engineering (1989), pp. 173–189Google Scholar
  19. 19.
    G.R. Liu, M.B. Liu, S. Li, Smoothed particle hydrodynamics: a mesh free particle method, in Computational Mechanics (2004)Google Scholar
  20. 20.
    T. Belytschko, Y.Y. Lu, L. Gu, Element free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229–256 (1994)CrossRefGoogle Scholar
  21. 21.
    I. Babuska, J.M. Melenk, The partition of unity methods. Int. J. Numer. Methods Eng. 40, 727–758 (1997)CrossRefGoogle Scholar
  22. 22.
    E. Onate, S. Idelsohn, O.C. Zienkiewicz, R.L. Taylor, A finite point method in computational mechanics. Application to convective transport and fluid flow. Int. J. Numer. Methods Eng. 39, 3839–3866 (1996)CrossRefGoogle Scholar
  23. 23.
    R. Jiwari, S. Singh, A. Kumar, Numerical simulation to capture the pattern formation of coupled reaction–diffusion models. Chaos Solitons Fractals 103, 422–439 (2017)CrossRefGoogle Scholar
  24. 24.
    A. Korkmaz, I. Dag, Polynomial based differential quadrature method for numerical solution of nonlinear Burgers equation. J. Frankl. Inst. 348(10), 28632875 (2011)CrossRefGoogle Scholar
  25. 25.
    R.C. Mittal, R. Jiwari, K.K. Sharma, A numerical scheme based on differential quadrature method to solve time dependent Burgers’ equation. Eng. Comput. 30(1), 117–131 (2013)CrossRefGoogle Scholar
  26. 26.
    R. Jiwari, R.C. Mittal, K.K. Sharma, A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers’ equation. Appl. Math. Comput. 219, 66806691 (2013)Google Scholar
  27. 27.
    R. Jiwari, A hybrid numerical scheme for the numerical solution of the Burgers equation. Comput. Phys. Commun. 188, 59–67 (2015)CrossRefGoogle Scholar
  28. 28.
    R. Jiwari, Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation. Comput. Phys. Commun. 183, 2413–2423 (2012)CrossRefGoogle Scholar
  29. 29.
    S. Tomasiello, A note on three numerical procedures to solve Volterra integro-differential equations in structural analysis. Comput. Math. Appl. 62, 3183–3193 (2011)CrossRefGoogle Scholar
  30. 30.
    S. Tomasiello, Some remarks on a new DQ-based method for solving a class of Volterra integro-differential equations. Appl. Math. Comput. 219, 399–407 (2012)Google Scholar
  31. 31.
    R. Jiwari, Lagrange interpolation and modified cubic B-spline differential quadrature methods for solving hyperbolic partial differential equations with Dirichlet and Neumann boundary conditions. Comput. Phys. Commun. 193, 55–65 (2015)CrossRefGoogle Scholar
  32. 32.
    G.F. Simmons, Differential Equations with Applications and Historical Notes, Mcgraw Hill Series in Mechanical Engineering, 2nd edn. (McGraw-Hill Education, New York, 2016)Google Scholar
  33. 33.
    R.C. Mittal, R. Rohila, Numerical simulation of reaction–diffusions ystems by modied cubic B-spline differential quadrature method. Chaos Solitons Fractals 92, 919 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia

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